Theorem f1orn 6828

Description: A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.)

Assertion

Ref	Expression
f1orn	⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹))

Proof of Theorem f1orn

Step	Hyp	Ref	Expression
1		dff1o2 6823	. 2 ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = ran 𝐹))
2		eqid 2735	. . 3 ⊢ ran 𝐹 = ran 𝐹
3		df-3an 1088	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = ran 𝐹) ↔ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹) ∧ ran 𝐹 = ran 𝐹))
4	2, 3	mpbiran2 710	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = ran 𝐹) ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹))
5	1, 4	bitri 275	1 ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ^◡ccnv 5653  ran crn 5655  Fun wfun 6525   Fn wfn 6526  –1-1-onto→wf1o 6530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-ex 1780  df-cleq 2727  df-ss 3943  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538

This theorem is referenced by:  f1f1orn  6829  infdifsn  9671  efopnlem2  26618  cycpmcl  33127